Discrete Mathematics 2010-2022

Additively indecomposable integers are a useful tool in the study of universal quadratic forms or the Pythagoras number in totally real number fields. In the case of real quadratic fields, they can be derived from the continued fraction of concrete quadratic irrationalities. Thus, it is natural to ask whether there exists analogous relation between indecomposable integers and multidimensional continued fractions. In this talk, we will focus on the Jacobi–Perron algorithm and discuss elements originating from expansions of concrete vectors for several families of cubic fields. This is joint work with Vítězslav Kala and Ester Sgallová.

We study normalized volumes of a family of polytopes associated with column-convex {0,1}-matrices. Such a family is a generalization of the family of consecutive coordinate polytopes, studied by Ayyer, Josuat-Vergès, and Ramassamy, which in turn generalizes a family of polytopes originally proposed by Stanley in EC1. We prove that a polytope associated with a column-convex {0,1}-matrix is integrally equivalent to a certain flow polytope. We use the recently developed machinery in the theory of volumes and lattice point enumeration of flow polytopes to find various expressions for the volume of these polytopes, providing new proofs and extending results of Ayyer, Josuat-Vergès, and Ramassamy. This is joint work with Chris Hanusa, Alejandro Morales, and Martha Yip.

10 Uhr: Prof. Dr. Carmelo Antonio FINOCCHIARO (University of Catania)
Titel: A Topological Insight on the Ideal Theory of Rings of Integer-Valued Polynomials

11 Uhr: Prof. Dr. Giulio PERUGINELLI (University of Padova)
Titel: The Ubiquity of Integer-Valued Polynomials: Polynomial Dedekind Domains

Gemeinsames Mittagessen

14 Uhr: Univ.-Prof. Dr. Daniel SMERTNIG (Universität Graz)
Titel: From Bass Rings to Monoids of Graph Agglomerations

15 Uhr: Dr. Roswitha RISSNER (Universität Klagenfurt)
Titel: Powers of Irreducibles in Rings of Integer-Valued Polynomials

Greedy algorithms are surprisingly effective in a wide range of optimization problems, though have only recently been considered as a possible way to find point configurations with low discrepancy or energy.
In this talk, we will discuss the performance of a greedy algorithm on the sphere to minimize the Riesz energies
$$ E_s(\{ z_1, ..., z_N\}) = \sum_{i \neq j} \frac{1}{s} \|x-y\|^{-s}$$
and the quadratic spherical cap discrepancy
$$ D_2(\{ z_1, ..., z_N\}) = \int_{-1}^{1} \int_{\mathbb{S}^d} \Big| \frac{\# ( C(x,t) \cap \{ z_1, ..., z_N\} )}{N} - \sigma(C(x,t)) \Big|^2 d\sigma(x) dt.$$
As an intermediate step, we study the maximal Riesz polarization on the sphere,
$$ \mathcal{P}_s( \mathbb{S}^d , N) = \sup_{\omega_N \subset \mathbb{S}^{d}, |\omega_N| = N} \; \inf_{x \in \mathbb{S}^d} \sum_{y \in \omega_N} \frac{1}{s} \|x - y \|^{-s}.$$
In an analogue of the connection between energy and best packings, as $s \rightarrow \infty$, optimal polarization configurations become best coverings. We show that that the first and second order asymptotics of optimal Riesz polarization nicely parallel those of energy minimization, and greedily generated point sets are nearly optimal for polarization.

The research in this presentation is joint work with Dmitriy Bilyk (University of Minnesota), Michelle Mastrianni (University of Minnesota), and Stefan Steinerberger (University of Washington).

The last decade has seen an ever growing research interest in the study of generalizations of spaces of “diagonal harmonics” for the symmetric group. Indeed, this bring into play natural interactions between Algebraic Combinatorics, Symmetric Function Theory, Representation Theory, Algebraic Geometry, Knot Theory, and Theoretical Physics. On the combinatorial side, the subject has evolved from the special case of “Classical Catalan Combinatorics” to the current all inclusive notion of “Triangular Catalan Combinatorics”, successively going through the contexts of “Rational Catalan Combinatorics” and then “Rectangular Catalan Combinatorics”, in increasing level of generality. Each of these has raised new questions in other fields. Just to illustrate, the rational case relates to calculation of the Khovanov-Rozansky homology of torus knots, whilst the rectangular case extends this to torus links.

We will describe interesting aspects of the combinatorics of the new triangular context, showing that it makes natural many relevant notions that previously appeared to be rather mysterious. We will also see that this context has nice closure properties that where lacking in previous ones. We will conclude with many open natural questions that remain to be explored.